A Numerical Study of Boson Star Binaries

Transcription

1 A Numerical Study of Boson Star Binaries Bruno C. Mundim with Matthew W. Choptuik (UBC) 12th Eastern Gravity Meeting Center for Computational Relativity and Gravitation Rochester Institute of Technology June 15, 2009

2 Outline General Motivation Matter Model: Scalar Field Conformally Flat Approximation (CFA) Motivation Formalism and Equations of motion Discretization Scheme and Numerical Techniques Initial Data Evolution Results Conclusion and Future Directions 1

3 General Motivation Why study compact binaries? One of most promising sources of gravitational waves. It is a good laboratory to study the phenomenology of strong gravitational fields. Why boson stars? Matter similarities: Fluid stars and Boson stars have some similarity concerning the way they are modelled, e.g. both can be parametrized by their central density ρ 0 and have qualitatively similar plots of total mass vs ρ 0. Then in the strong field regime for the compact binary system the dynamics may not depend sensitively on the details of the model. Inspiral phases: Plunge and merge phase of the inspiral of compact objects is characterized by a strong dynamical gravitational field. In this regime gross features of fluid and boson stars dynamics may be similar. Since the details of the dynamics of the stars (e.g. shocks) tend not to be important gravitationally, boson star binaries may provide some insight into NS binaries. 2

4 Matter Model: Scalar Field Star-like solutions: A massive complex field is chosen as matter source because it is a simple type of matter that allows a star-like solution and because there will be no problems with shocks, low density regions, ultrarelativistic flows, etc in the evolution of this kind of matter as opposed to fluids. Static spacetimes: Complex scalar fields allow the construction of static spacetimes. The matter content is then described by Φ = φ 1 + iφ 2, where φ 1 and φ 2 are real-valued. Equations of motion: Klein-Gordon equation: φ A m 2 φ A = 0, A = 1, 2. (1) Hamiltonian Formulation: In terms of the conjugate momentum field Π A : t φ A = α2 g Π A + β i i φ A, (2) t Π A = i (β i Π A ) + i ( gγ ij j φ A ) gm 2 φ A. (3) 3

5 Conformally Flat Approximation (CFA) Motivation Facts and assumptions: Full 3D Einstein equations are very complex and computationally expensive to solve. Gravitational radiation is small in most systems studied so far. Heuristic assumption that the dynamical degrees of freedom of the gravitational fields, i.e. the gravitational radiation, play a small role in at least some phases of the strong field interaction of a merging binary. An approximation candidate: CFA effectively eliminates the two dynamical degrees of freedom, simplifies the equations and allows a fully constrained evolution. CFA allows us to investigate the same kind of phenomena observed in the full relativistic case, such as the description of compact objects and the dynamics of their interaction; black hole formation; critical phenomena. 4

6 Conformally Flat Approximation (CFA) Formalism Einstein field equations cast into 3+1/ADM form. The CFA prescribes a conformally flat spatial metric at all times. Introduce a flat metric f ij as a base / background metric: γ ij = ψ 4 f ij, (4) where the conformal factor ψ is a positive scalar function describing the ratio between the scale of distance in the curved space and flat space (f ij δ ij in cartesian coordinates). Maximum slicing condition is used to fix the time coordinate: K i i = 0, t K i i = 0. (5) In this approximation all of the geometric variables can be computed from the constraints as well as from a specific choice of coordinates. 5

7 Conformally Flat Approximation (CFA) Slicing Condition Gives an elliptic equation for the lapse function α: 2 α = 2 ψ ψ α + αψ 4 ( K ij K ij + 4π (ρ + S) ). (6) Hamiltonian Constraint Gives an elliptic equation for the conformal factor ψ: 2 ψ = ψ5 8 ( Kij K ij + 16πρ ). (7) Momentum Constraints Given elliptic equations for the shift vector components β i : 2 β j = 1 3ˆγij i ( β ) ( )] [ ψ + αψ 4 16πJ j 6 i [ln ˆγ ik k β j α +ˆγ jk k β i 2 3ˆγij ( β ) ]. (8) 6

8 Conformally Flat Approximation (CFA) Note that K ij K ij can also be expressed in terms of the flat operators. It ends up being expressed as flat derivatives of the shift vector: K ij K ij = 1 2α 2 (ˆγ knˆγ ml ˆDm β k ˆDl β n + ˆD m β l ˆDl β m 23 ) ˆD l β l ˆDk β k. (9) 7

9 Conformally Flat Approximation (CFA) Then the following set of functions completely characterize the geometry at each time slice: α = α(t, r), ψ = ψ(t, r), β i = β i (t, r), (10) where r depends on the coordinate choice for the spatial hypersurface. The solution of the gravitational system under CFA and maximal slicing condition can be summarized as: Specify initial conditions for the complex scalar field. Solve the elliptic equations for the geometric quantities on the initial slice. Update the matter field values to the next slice using their equation of motion. For the new configuration of matter fields, re-solve the elliptic equations for the geometric variables and again allow the matter fields to react and evolve to the next slice and so on. 8

10 Conformally Flat Approximation (CFA) Discretization Scheme: Lu f = 0 L h u h f h = 0. (11) For hyperbolic operators L: second order accurate Crank-Nicholson scheme. For elliptic operators L: second order accurate centred finite difference operators. Dirichlet Boundary conditions applied. Numerical Techniques: pointwise Newton-Gauss-Seidel (NGS) iterative technique was used to solve the finite difference equations originated from the hyperbolic set of equations. Full Approximation Storage (FAS) multigrid algorithm was applied on the discrete version of the elliptic set of equations. NGS is used in this context as a smoother of the solution error. 9

11 Initial Data Static spherically symmetric ansatz: φ(t, r) = φ 0 (r)e iωt. 10

12 Initial Data Family of static spherically symmetric solutions: 11

13 Evolution Results Remarks: ρ φ(t, r) 2 = φ 0 (r) 2, and Planck units are adopted, i.e. G = c = = 1. Results summary: Initial data: Each boson star is modelled as a static, spherically symmetric solution of the Einstein-Klein-Gordon system. They each have a central scalar field value of φ 0 (0) = 0.02, that corresponds to a boson star with radius of R and ADM mass of M ADM Each solution is then superposed and boosted in opposite directions (along x axis). Evolution: Orbital motion and interrupted orbits: The stars lie along the y axis with a coordinate separation between their centers of 40. Three distinct cases studied corresponding to different initial velocities: v x = 0.09: two orbital periods. v x = 0.07: rotating boson star as final merger. v x = 0.05: possible black-hole formation. Head-on collision: Stars along x axis with coordinate separation of 50: v x = 0.4: solitonic behaviour. 12

14 Evolution Results Orbital Dynamics: 2 boson stars - v x = Z = 0 slice for φ Z = 0 slice for α φ 0 : Physical coordinate domain: 120 per edge. Physical time: t = Simulation parameters: Courant factor λ = 0.4; Grid size: ; 2.4GHz Dual-Core AMD Opteron CPU time: 285 hours (12 days). 13

15 Evolution Results Orbital Dynamics: 2 boson stars - v x = Z = 0 slice for φ Z = 0 slice for α φ 0 : Physical coordinate domain: 120 per edge. Physical time: t = Simulation parameters: Courant factor λ = 0.4; Grid size: ; 2.4GHz Dual-Core AMD Opteron CPU time: 158 hours (6.5 days). 14

16 Evolution Results Orbital Dynamics: 2 boson stars - v x = Z = 0 slice for φ Z = 0 slice for α φ 0 : Physical coordinate domain: 120 per edge. Physical time: t = Simulation parameters: Courant factor λ = 0.4; Grid size: ; 2.4GHz Dual-Core AMD Opteron CPU time: 115 hours (4.5 days). 15

17 Evolution Results Head-on collision: 2 boson stars - v x = 0.4. Z = 0 slice for φ Z = 0 slice for α φ 0 : Physical coordinate domain: [ 50, 50, 25, 25, 25, 25]. Total physical time: t = 140. Simulation parameters: λ = 0.4; Grid size: [N x, N y, N z ] = [129, 65, 65]; 16

18 Conclusion and Future Directions We were able to probe a few outcomes of the orbital dynamics of boson stars within the CFA. At least qualitatively, we observed a few characteristic phenomena of the fully relativistic case, such as orbital precession and scalar matter solitonic behaviour. These results are quite promising and suggest that, with enhancements such as the incorporation of AMR techniques and parallel execution capabilities, this code will be a powerful tool for investigating the strong gravity effects in the interaction of boson stars. We hope, in the future, to be able to calibrate the CFA fidelity to the fully general relativistic case and use this code to survey the parameter space of the orbital dynamics. 17

19 Appendix A - CFA equations of motion 3d Cartesian Coordinates t φ A = α ψ 6Π A + β i i φ A (12) t Π A = x ( β x Π A + αψ 2 x φ A ) + y ( β y Π A + αψ 2 y φ A ) ( + z β z Π A + αψ 2 ) z φ A αψ 6 du(φ2 0) dφ 2 φ A 0 (13) 2 α x 2+ 2 α y α z 2 = 2 ψ [ ψ α x x + ψ α y y + ψ z ] α +αψ 4 ( K ij K ij + 4π (ρ + S) ) z (14) 2 ψ x ψ y ψ z 2 = ψ5 8 ( Kij K ij + 16πρ ) (15) 18

20 Appendix A - CFA equations of motion x component of the shift vector in cartesian coordinates 2 β x x β x y β x z 2 = 1 ( ) β x 3 x x + βy y + βz + αψ 4 16πJ x z [ ( )] [ ψ 6 4 β x ln x α 3 x 2 ( )] β y 3 y + βz z [ ( )] [ ] ψ 6 β x ln y α y + βy x [ ( )] [ ] ψ 6 β x ln z α z + βz (16) x K ij K ij in 3d cartesian [ ( β ) coordinates K ij K ij = 1 x 2 ( ) 2α 2 x + β x 2 ( ) y + β x 2 ( ) z + β y 2 ( ) x + β y 2 ( ) y + β y 2 z ( ) + β z 2 ( ) x + β z 2 ( ) y + β z 2 ( ) z + x β x x + βy y + βz z β x ( ) ( ) + y β x x + βy y + βz z β y + z β x x + βy y + βz z β z ( ) ] 2 β x 2 3 x + βx x + βx x (17) 19

21 Appendix B: Boson Stars in Spherical Symmetry Spherically Symmetric Spacetime (SS): ds 2 = ( α 2 + a 2 β 2) dt 2 + 2a 2 β dtdr + a 2 dr 2 + r 2 b 2 dω 2, (18) Hamiltonian constraint: { [(rb) 2 ] + 1 arb a rb [ (rb a (rb) ) a]} + 4K r rk θ θ + 2K θ θ2 = [ Φ 2 + Π 2 ] 8π a 2 + m 2 φ 2 (19) Momentum constraint: K θ θ + (rb) rb where the auxiliary field variables were defined as: ( K θ θ K r r) = 2π a (Π Φ + ΠΦ ). (20) Φ φ, (21) Π a α ( φ βφ ), (22) 20

22 Evolution equations Boson Stars in Spherical Symmetry ȧ = αak r r + (aβ) (23) ḃ = αbk θ θ + β r (rb). K r r = βk r r 1 ( ) { α + α 2 a a arb (24) [ ] (rb) [ ] } 2 Φ + KK r 2 r 4π a a 2 + m 2 φ 2 (25) K θ θ = βk θ θ + α (rb) 2 1 a(rb) 2 Field evolution equations [ αrb a (rb) ] + α ( KK θ θ 4πm 2 φ 2) (26) φ = α a Π + βφ (27) Φ = Π = (βφ + α a Π ) 1 [ ( (rb) 2 (rb) 2 βπ + α )] a Φ αam 2 φ + 2 (28) ] [αk θθ β (rb) Π (29) rb 21

23 Appendix B: Boson Stars in Spherical Symmetry Maximal-isotropic coordinates Maximal slicing condition K Ki i = 0 K(t, r) = 0 (30) Isotropic condition α + 2 rψ 2 d dr 2 ( r 2 ψ 2) α + a = b ψ(t, r) 2 (31) They fix the lapse and shift (equivalent of fixing the coordinate system) [ 4πψ 4 m 2 φ 2 8π Π 2 3 ] ( ψ 2 K r ) 2 r α = 0 (32) 2 Constraint equations 3 d ψ 5 dr 3 ( r 2dψ ) dr r ( ) β = 3 r 2 αkr r (33) + 3 ( Φ 16 Kr r Π 2 ) = π ψ 4 + m 2 φ 2 (34) K r r + 3 (rψ2 ) rψ 2 Kr r = 4π ψ 2 (Π Φ + ΠΦ ) (35) 22

24 Appendix B: Boson Stars in Spherical Symmetry Complex-scalar field evolution equations φ = α ψ2π + βφ Φ = (βφ + αψ ) 2Π Π = 3 ψ 4 d dr 3 [r 2 ψ 4 (βπ + αψ 2Φ )] (36) (37) αψ 2 m 2 φ [αk rr + 2β (rψ2 ) ] rψ 2 Π (38) 23

25 Appendix B: Boson Stars in Spherical Symmetry These equations were coded using RNPL and tested for a gaussian pulse as initial data. 24

26 Appendix B: Boson Stars in Spherical Symmetry Initial Value Problem We are interested in generating static solutions of the Einstein- Klein-Gordon system There is no regular, time-independent configuration for complex scalar fields but one can construct harmonic time-dependence that produce time-independ ent metric We adopt the following ansatz for boson stars in spherical symmetry in order to produce a static spacetime: φ(t, r) = φ 0 (r) e iωt, β = 0 (39) where the last condition comes from the demand of a static timelike Killing vector field. Polar-Areal coordinates K = K r r b = 1 (40) Generalization of the usual Schwarzschild coordinates to time-dependent, spherically symmetric spacetimes. Easier to generate the initial data solution 25

27 Appendix B: Boson Stars in Spherical Symmetry The line element ds 2 = α 2 dt 2 + a 2 dr 2 + r 2 dω 2. (41) The equations of motions are cast in a system of ODEs. It becomes an eigenvalue problem with eigenvalue ω = ω(φ 0 (0)) a = 1 { a ( 1 a 2 ) ) ]} + 4πra [φ 2 a (m ω2 2 r α 2 + Φ 2 α = α { a 2 ( ) ]} 1 ω + 4πr [a 2 φ r α 2 m2 + Φ 2 φ = Φ Φ = ( 1 + a 2 4πr 2 a 2 m 2 φ 2) Φ r (42) (43) (44) ( ) ω 2 α 2 m2 φa 2 (45) 26

28 Appendix B: Boson Stars in Spherical Symmetry Field configuration and its aspect mass function for φ 0 (0) = Its eigenvalue was shooted to be ω = Note its exponentially decaying tail as opposed to the sharp edge ones for its fluids counterparts 27

29 Appendix B: Boson Stars in Spherical Symmetry The ADM mass as a function of the central density and the radius of the star as a function of ADM mass. Note their similarity to the fluid stars 28